2014 Lectures on Computational and Applied Mathematics
(a) Thomas Hou, Caltech, USA
(b) Quoqiang Shi, Tsinghua University, China
(a) Data Driven Time-Frequency Analysis
(b) Data-Driven Stochastic Methods for Solving Stochastic PDEs
(c) Recent Progress on the Search of 3D Euler Singularities
(d) Finite Integral Method: A numerical approach for Poisson equation on point clouds
March 20(Thursday), 2014 PM 1:30-4:00
March 21, 2014 (Friday) AM 9:30-12:00 & PM1:30-4:00
Place: Hong-Jing Building M107
(1). Data Driven Time-Frequency Analysis
In this talk, we review some recent progress on our
data-driven time-frequency analysis method for studying trend and
instantaneous frequency of nonlinear and non-stationary data. This method is inspired by the Empirical Mode Decomposition method (EMD) and the recently developed compressed sensing theory. The main idea is to look for the sparsest representation of multiscale data within the largest possible dictionary consisting of intrinsic mode functions. This problem can be formulated as a nonlinear optimization problem. In order to solve this optimization problem, we propose a nonlinear matching pursuit method by generalizing the classical matching pursuit. One important advantage of this nonlinear matching pursuit method is it can be implemented very efficiently and is very stable to noise. Further, we provide a convergence analysis of our nonlinear matching pursuit method under certain scale separation assumptions. Extensive numerical examples will be given to demonstrate the robustness of our method and comparison will be made with the EMD/EEMD method. We also apply our method to study data without scale separation, data with intra-wave frequency modulation, and data with incomplete or under-sampled data.
(2). Data-Driven Stochastic Methods for Solving Stochastic PDEs.
Uncertainty arises in many complex real-world problems of
scientific and engineering interests. Many of these problems involve
multiple scales in both space and time, which may vary in several orders. We have recently developed an effective dynamically bi-orthogonal method (DyBO) for solving time-dependent stochastic PDEs. The main idea is to construct the most compact representation of the stochastic PDEs by tracking the Karhunen-Loeve expansion dynamically. We achieve this by deriving an equivalent set of evolution systems that govern the stochastic and deterministic basis without the need to form the covariance matrix or to compute its eigen-decomposition. Unlike other reduced model methods, our method constructs the reduced stochastic basis on-the-fly. We also propose an adaptive strategy to dynamically remove or add modes, and develop an efficient parallel algorithm for our method. Applications to solve 2D incompressible Navier-Stokes equations and the Boussinesq approximation with Brownian motion forcing demonstrate the effectiveness of the DyBO method. We will also introduce a data-driven method for solving stochastic elliptic problems with multiscale coefficient in a multi-query setting. A multiscale multilevel Monte-Carlo method has been developed to handle stochastic problems with high input dimensions. Our numerical experiments show that these methods could offer considerable computational saving over existing methods.
(3). Recent Progress on the Search of 3D Euler Singularities.
Whether the 3D incompressible Euler or Navier-Stokes
equations can develop a finite time singularity from smooth initial data
with finite energy is one of the most challenging questions in applied
mathematics and fluid dynamics. We review some recent theoretical and computational studies of the 3D Euler equations which show that there is a subtle dynamic depletion of nonlinear vortex stretching due to local geometric regularity of vortex filaments. Our study suggests that the convection term could have a nonlinear stabilizing effect for certain
flow geometry. This is demonstrated through two reduced models of the 3D incompressible Navier-Stokes equations. The first model is a new exact 1D model for the axisymmetric Navier-Stokes equation along the symmetry axis. We show that local flattening of the vortex structure and the effect of convection could lead to dynamic depletion of the vortex
stretching term. In the second model, we derive a 3D model of the
axisymmetric Navier-Stokes equation by removing the convection term from the reformulated Navier-Stokes equations. We show both numerically and analytically that the solution of this 3D model develops a finite time singularity from smooth initial data with finite energy. Finally we present a new class of solutions to the 3D Euler equations which could lead to a strong nonlinear alignment in the vortex stretching term and have the potential to develop a finite time singularity.
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